### Bi-College Math Colloquium

Monday, November 1, 2021 - 4:00pm

#### Renee Bell

University of Pennsylvania

Galois theory describes the rich connection between field theory and group theory. Similarly, the fundamental group in topology connects group theory to the study of topological spaces. In this talk, we formalize this analogy with the étale fundamental group π1(X). Over fields of characteristic zero, π1(X) closely resembles its topological analogue, but in characteristic p, dramatic differences and new phenomena have inspired many conjectures. Let k be an algebraically closed field of characteristic p and let X be the projective line over k with three points removed. In joint work with Booher, Chen, and Liu, we show that for each prime p≥5, there are families of tamely ramified covers with monodromy the symmetric group Sn or alternating group An for infinitely many n, producing these covers from moduli spaces of elliptic curves.